Linear Mixed Models
Researchers study 4 knee braces: Control, Air Armor 2, Breg, Don Joy
10 high school running backs with previous ACL injury participate. Each player: Wears all 4 braces, Order randomized
Time to complete 40-yard agility course recorded
Do we care only about these 10 running backs?
| Source of Variation | DF |
|---|---|
Linear Mixed Models (LMM)
Mixed models come about when one factor is fixed and one factor is random. A mixed model should be used to study the effects of one factor on the population mean and the effects of another factor on the population variance.
\[y_{ij} = \mu + \tau_i + p_j + \epsilon_{ij} \text{ with } p_j \sim \text{iid } N(0,\sigma_{blk}^2) \text{ and } \epsilon_{ij} \sim \text{iid } N(0,\sigma_\epsilon^2)\]
for \(i=1,2,3; j=1,2,…,10\)
where:
Note
# A tibble: 12 × 3
RunningBack KneeBrace AgilityTime
<fct> <fct> <dbl>
1 1 Control 19.1
2 1 AirArmor 18
3 1 Breg 18.0
4 1 DonJoy 17.9
5 2 Control 19
6 2 AirArmor 18.4
7 2 Breg 18.2
8 2 DonJoy 17.7
9 3 Control 19.9
10 3 AirArmor 18.7
11 3 Breg 18.6
12 3 DonJoy 18.7
Linear mixed model fit by REML ['lmerMod']
Formula: AgilityTime ~ KneeBrace + (1 | RunningBack)
Data: kneebrace_data
REML criterion at convergence: 1.2
Scaled residuals:
Min 1Q Median 3Q Max
-1.72607 -0.47036 -0.03709 0.53329 1.71255
Random effects:
Groups Name Variance Std.Dev.
RunningBack (Intercept) 0.15453 0.3931
Residual 0.01965 0.1402
Number of obs: 40, groups: RunningBack, 10
Fixed effects:
Estimate Std. Error t value
(Intercept) 18.24200 0.13198 138.22
KneeBraceBreg -0.01000 0.06269 -0.16
KneeBraceControl 0.97500 0.06269 15.55
KneeBraceDonJoy -0.16800 0.06269 -2.68
Correlation of Fixed Effects:
(Intr) KnBrcB KnBrcC
KneeBracBrg -0.237
KneBrcCntrl -0.237 0.500
KneeBrcDnJy -0.237 0.500 0.500
Analyze > Fit Model
\[H_0: \tau_1=\tau_2=\tau_3=\tau_4 \text{ vs } H_A: \text{ at least one } \tau_i \text{ differs}\]
Recall \(y_{ij} = \mu + \tau_i + p_j + \epsilon_{ij}\)
In the mixed model:
Thus, the estimated mean and variance are given by:
KneeBrace emmean SE df lower.CL upper.CL .group
Control 19.2 0.132 10.7 18.8 19.6 A
AirArmor 18.2 0.132 10.7 17.8 18.6 B
Breg 18.2 0.132 10.7 17.8 18.6 B
DonJoy 18.1 0.132 10.7 17.7 18.5 B
Degrees-of-freedom method: kenward-roger
Confidence level used: 0.95
Conf-level adjustment: sidak method for 4 estimates
P value adjustment: tukey method for comparing a family of 4 estimates
significance level used: alpha = 0.05
NOTE: If two or more means share the same grouping symbol,
then we cannot show them to be different.
But we also did not show them to be the same.
Treating blocks as random (instead of fixed) (1) Increases SE of treatment means (2) Does NOT change SE of treatment differences
Running back Fixed (LM)
KneeBrace emmean SE df lower.CL upper.CL
AirArmor 18.24 0.0443 27 18.15 18.33
Breg 18.23 0.0443 27 18.14 18.32
Control 19.22 0.0443 27 19.13 19.31
DonJoy 18.07 0.0443 27 17.98 18.16
Results are averaged over the levels of: RunningBack
Confidence level used: 0.95
contrast estimate SE df t.ratio p.value
AirArmor - Breg 0.010 0.0627 27 0.160 0.9985
AirArmor - Control -0.975 0.0627 27 -15.553 <0.0001
AirArmor - DonJoy 0.168 0.0627 27 2.680 0.0565
Breg - Control -0.985 0.0627 27 -15.713 <0.0001
Breg - DonJoy 0.158 0.0627 27 2.520 0.0792
Control - DonJoy 1.143 0.0627 27 18.233 <0.0001
Results are averaged over the levels of: RunningBack
P value adjustment: tukey method for comparing a family of 4 estimates
Running back Random (LMM)
KneeBrace emmean SE df lower.CL upper.CL
AirArmor 18.2 0.132 10.7 18.0 18.5
Breg 18.2 0.132 10.7 17.9 18.5
Control 19.2 0.132 10.7 18.9 19.5
DonJoy 18.1 0.132 10.7 17.8 18.4
Degrees-of-freedom method: kenward-roger
Confidence level used: 0.95
contrast estimate SE df t.ratio p.value
AirArmor - Breg 0.010 0.0627 27 0.160 0.9985
AirArmor - Control -0.975 0.0627 27 -15.553 <0.0001
AirArmor - DonJoy 0.168 0.0627 27 2.680 0.0565
Breg - Control -0.985 0.0627 27 -15.713 <0.0001
Breg - DonJoy 0.158 0.0627 27 2.520 0.0792
Control - DonJoy 1.143 0.0627 27 18.233 <0.0001
Degrees-of-freedom method: kenward-roger
P value adjustment: tukey method for comparing a family of 4 estimates